How to put an object at the outer end of each branch of a sapling?
With help of lemon's initial comment, which pointed me in the right direction, I arrived at this solution.  

Once you happy with the shape of your tree, select it and hit AltC then M to convert it to a mesh, then Tab into Edit mode. Hit A once or twice if needed to select all.
  

Hit CtrlV then D to remove doubles. Reduce the Merge Distance in the options panel in the lower left corner. Set it as low as it will go.
  

Select the end vertex of one branch, the hit ShiftG then O to select every vertex that has the same number of edges connected to it as the already selected one.
  

Instead of creating a vertex group, I duplicated the selected vertices (ShiftD) and then separated them into a new object (P then S). Now Tab out of Edit mode and hide the complete tree by clicking the eye next to its name in the Outliner.
  

Select the other "tree", which is just a cloud of unconnected vertices. You should see something like this.
  

Add a sphere (or whatever object you want at the ends of the branches). Hold Shift and select the vertex cloud. Now the sphere and the vertex cloud should both be selected, with the vertex cloud being the active object. Hit CtrlP then O. The vertex cloud is now the parent of the sphere. Select the vertex cloud and go into the Object tab of the Properties panel. Under Duplication, click Verts.
  

Scale the sphere (select it and hit S, drag the mouse until things look good, then hit Enter). You should now have something similar to this.
  

And if you unhide the original tree, by again clicking the eye next to its name in the Outliner, it should look like this.
  

The spheres in the tree are now just copies of the original sphere. The original sphere and the vertex cloud need to be left. They will be visible in the 3D view, but not in the render.
The duplicates can be made real, if you want or need to, by selecting the vertex cloud (i.e. the parent of the sphere which is being duplicated) and hitting ShiftCtrlA. After that you can delete the vertex cloud and the original sphere.